Comparison of the 3D Numerical Schemes for Solving Curvature Driven Level Set Equation Based on Discrete Duality Finite Volumes

Dana Kotorová

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2014)

  • Volume: 53, Issue: 2, page 71-83
  • ISSN: 0231-9721

Abstract

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In this work we describe two schemes for solving level set equation in 3D with a method based on finite volumes. These schemes use the so-called dual volumes as in [Coudiére, Y., Hubert, F.: A 3D discrete duality finite volume method for nonlinear elliptic equations Algoritmy 2009 (2009), 51–60.], [Hermeline, F.: A finite volume method for approximating 3D diffusion operators on general meshes Journal of Computational Physics 228, 16 (2009), 5763–5786.], where they are used for the nonlinear elliptic equations. We describe these schemes theoretically and also compare results of the numerical experiments based on exact solution using proposed schemes.

How to cite

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Kotorová, Dana. "Comparison of the 3D Numerical Schemes for Solving Curvature Driven Level Set Equation Based on Discrete Duality Finite Volumes." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 53.2 (2014): 71-83. <http://eudml.org/doc/262165>.

@article{Kotorová2014,
abstract = {In this work we describe two schemes for solving level set equation in 3D with a method based on finite volumes. These schemes use the so-called dual volumes as in [Coudiére, Y., Hubert, F.: A 3D discrete duality finite volume method for nonlinear elliptic equations Algoritmy 2009 (2009), 51–60.], [Hermeline, F.: A finite volume method for approximating 3D diffusion operators on general meshes Journal of Computational Physics 228, 16 (2009), 5763–5786.], where they are used for the nonlinear elliptic equations. We describe these schemes theoretically and also compare results of the numerical experiments based on exact solution using proposed schemes.},
author = {Kotorová, Dana},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Mean curvature flow; level set equation; numerical solution; semi-implicit scheme; discrete duality finite volume method (DDFV); mean curvature flow; level set equation; semi-implicit scheme; discrete duality finite volume method; numerical experiments},
language = {eng},
number = {2},
pages = {71-83},
publisher = {Palacký University Olomouc},
title = {Comparison of the 3D Numerical Schemes for Solving Curvature Driven Level Set Equation Based on Discrete Duality Finite Volumes},
url = {http://eudml.org/doc/262165},
volume = {53},
year = {2014},
}

TY - JOUR
AU - Kotorová, Dana
TI - Comparison of the 3D Numerical Schemes for Solving Curvature Driven Level Set Equation Based on Discrete Duality Finite Volumes
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2014
PB - Palacký University Olomouc
VL - 53
IS - 2
SP - 71
EP - 83
AB - In this work we describe two schemes for solving level set equation in 3D with a method based on finite volumes. These schemes use the so-called dual volumes as in [Coudiére, Y., Hubert, F.: A 3D discrete duality finite volume method for nonlinear elliptic equations Algoritmy 2009 (2009), 51–60.], [Hermeline, F.: A finite volume method for approximating 3D diffusion operators on general meshes Journal of Computational Physics 228, 16 (2009), 5763–5786.], where they are used for the nonlinear elliptic equations. We describe these schemes theoretically and also compare results of the numerical experiments based on exact solution using proposed schemes.
LA - eng
KW - Mean curvature flow; level set equation; numerical solution; semi-implicit scheme; discrete duality finite volume method (DDFV); mean curvature flow; level set equation; semi-implicit scheme; discrete duality finite volume method; numerical experiments
UR - http://eudml.org/doc/262165
ER -

References

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  1. Andreianov, B., Bendahmare, M., Karlsen, K. H., A Gradient Reconstruction Formula for Finite Volume Schemes and Discrete Duality, In: Finite Volume For Complex Applications, Problems And Perspectives. 5th International Conference, Wiley, London, 2008, 161–168. (2008) MR2451403
  2. Andreianov, B., Boyer, F., Hubert, F., 10.1002/num.20170, Numerical Methods PDE 23, 1 (2007), 145–195. (2007) Zbl1111.65101MR2275464DOI10.1002/num.20170
  3. Coudiére, Y., Hubert, F., A 3D discrete duality finite volume method for nonlinear elliptic equations, Algoritmy 2009 (2009), 51–60. (2009) Zbl1171.65441
  4. Evans, L. C., Spruck, J., Motion of the level sets by mean curvature I, J. Differential Geometry 3 (1991), 635–681. (1991) MR1100206
  5. Eymard, R., Gallouë, T., Herbin, R., Finite volume methods, Handbook of Numerical Analysis (Ph., Ciarlet, J. L., Lions, eds.), 3 (2000), 713–1018. (2000) MR1804748
  6. Handlovičová, A., Kotorová, D., Stability of the semi-implicit discrete duality finite volume scheme for the curvature driven level set equation in 2D, Tatra Mountains Mathematical Publications, accepted. 
  7. Hermeline, F., 10.1016/j.jcp.2009.05.002, Journal of Computational Physics 228, 16 (2009), 5763–5786. (2009) Zbl1168.76340MR2542915DOI10.1016/j.jcp.2009.05.002
  8. Kotorová, D., Discrete duality finite volume scheme for the curvature-driven level set equation, Acta Polytechnica Hungarica 8, 3 (2011), 7–12. (2011) 
  9. Kotorová, D., Discrete duality finite volume scheme for the curvature driven level set equation in 3D, In: Advances in architectural, civil and environmental engineering [electronic source]: 22nd Annual PhD Student Conference, Nakl. STU, Bratislava, 2012, 33–39. (2012) 
  10. Kotorová, D., 3D numerical schemes for the level set equation based on discrete duality finite volumes, to appear. 
  11. Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science, Cambridge University Press, New York, 1999. (1999) MR1700751

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